CH. 5FACTORIAL DESIGN
I N T R O D U C T I O N
• Factorial design is an important method to determine the effects of MULTIPLE variables on a response/output.
• Traditionally, experiments are designed to determine the effect of ONE variable upon ONE response/output.
https://controls.engin.umich.edu/wiki/index.php/Design_of_experiments_via_factorial_designs
Tensile Strengt
h
Cooling Rate
%CAlloying Elemen
ts
Heating Temp
Heat Transfer Rate
Temp Diff
Physical Prop.
Fluid Velocity
Heat Transfer Area
Which variable is more important than others?
• R.A. Fisher showed that there are advantages by combining the study of multiple variables in the same factorial experiment. Factorial design can reduce the number of experiments one has to perform by studying multiple factors simultaneously.
• Additionally, it can be used to find both main effects (from each independent factor) and interaction effects (when both factors must be used to explain the outcome).
https://controls.engin.umich.edu/wiki/index.php/Design_of_experiments_via_factorial_designs
• Factorial design is a useful method to design experiments in both laboratory and industrial settings.
• Because factorial design can lead to a large number of trials, which can become expensive and time-consuming, factorial design is best used for a small number of variables with few states (1 to 3).
https://controls.engin.umich.edu/wiki/index.php/Design_of_experiments_via_factorial_designs
A 23 FACTORIAL DESIGN: PILOT PLANT INVESTIGATION
• This experiment employed a 23 factorial experimental design with two quantitative factors—temperature T and concentration C—and a single qualitative factor—type of catalyst K.
• Each data value recorded is for the response yield y averaged over two duplicate runs.
CALCULATION OF MAIN EFFECTS
CALCULATION OF MAIN EFFECTS
CALCULATION OF MAIN EFFECTS
INTERACTION EFFECTS
I N T E R A C T I O N E F F E C T S
I N T E R A C T I O N E F F E C T S
A L G O R I T M A YAT E S
• To calculate – The main effects (e.g. main effect 1, main effect 2,
etc.)– The interaction effects (12, 13, 23, 123, etc.)
STANDARD ERROR FOR EFFECTS
effect
effect
VSE
sN
V
24
Repl-1 Repl-2 Varians59 61 274 70 850 58 3269 67 250 54 881 85 846 44 279 81 2
SUM-s^2= 64Average-SUM-s^2= 8
V(effect)= 2SE = 1,41
Yield
S TA N D A R D E R R O R
• Standard error = standard of deviation.
nsSE2
15,1323,136628,472
nsSE
http://hatta2stat.wordpress.com/2011/05/21/standar-error/
EXAMPLE-2
EXAMPLE-3
FACTORIAL DESIGN WITH REPLICATION RUNS
• Checked by ANOVA.• Comparing variation between treatment (ST )
and variation within treatment (SR ).• Conclusion..?
12
1
2
kSs
yynS
TT
k
tttT
COMPARING TWO TREATMENTS
• Checked by Fisher Test and Student Test.• Conclusion..?
25,972
min
2max ss
21
1121
5,97
nnsxxt
tt